generalized Farkas lemma
Farkas’ Lemma of convex optimization and linear programming can be formulated for topological vector spaces^{}.
The more abstract version of Farkas’ Lemma is useful for understanding the essence of the usual version of the lemma proven for matrices, and of course, for solving optimization problems in infinitedimensional spaces.
The key insight is that the notion of linear inequalities in a finite number of real variables can be generalized to abstract linear spaces by the concept of a cone.
0.1 Formal statements
Farkas’ Lemma may be stated in the several equivalent^{} ways. Theorem 1 is conceptually the simplest, but Theorem 2 and 3 are more convenient for applications.
Theorem 1.
Let $X$ be a real vector spaces, and ${X}^{\mathrm{\prime}}$ be a subspace^{} of linear functionals^{} on $X$ that separate points. Impose on $X$ the weak topology generated by ${X}^{\mathrm{\prime}}$.
Given $x\mathrm{\in}X$, and a weaklyclosed convex cone $K\mathrm{\subseteq}X$, the following are equivalent:
Proof.
The equivalence of conditions (a) and (c) is a fundamental property of the anticone, while condition (b) is merely a rephrasal of condition (c). ∎
Theorem 2 is a version of Theorem 1 where the vector space and its dual space^{} switch roles.
Theorem 2.
Let $X$ be a real vector space, and ${X}^{\mathrm{\prime}}$ be a subspace of linear functionals on $X$ that separate points. Impose on ${X}^{\mathrm{\prime}}$ the weak* topology generated by $X$.
Given a functional^{} $f\mathrm{\in}{X}^{\mathrm{\prime}}$, and a weak* closed convex cone $K\mathrm{\subseteq}{X}^{\mathrm{\prime}}$, the following are equivalent:

(a)
$f\in K$.

(b)
$\{x\in X:f(x)\ge 0\}\supseteq {\bigcap}_{\varphi \in K}\{x\in X:\varphi (x)\ge 0\}$.
Proof.
Make the substitutions $X\to {X}^{\prime}$, ${X}^{\prime}\to X$, $x\to f$ and $K\to K$ in Theorem 1. ∎
Theorem 3 incorporates inequalities^{} defined by linear mappings; such linear mappings are the analogues to the matrices involved in the finitedimensional version of Farkas’ Lemma.
Theorem 3.
Let $X$ and $Y$ be real vector spaces, with corresponding spaces of linear functionals ${X}^{\mathrm{\prime}}$ and ${Y}^{\mathrm{\prime}}$ that separate points. Have ${X}^{\mathrm{\prime}}$ and ${Y}^{\mathrm{\prime}}$ generate the weak topology for $X$ and $Y$ respectively.
Given $y\mathrm{\in}Y$, a linear mapping $T\mathrm{:}X\mathrm{\to}Y$, and a subset $K\mathrm{\subseteq}X$ such that $T\mathit{}\mathrm{(}K\mathrm{)}$ is a weaklyclosed convex cone, the following are equivalent:

(a)
The linear equation $Tx=y$ has a solution $x\in K$.

(b)
If $\psi \in {Y}^{\prime}$ satisfies $\psi (Tx)\ge 0$ for all $x\in K$, then $\psi (y)\ge 0$.

(c)
If $\psi \in {Y}^{\prime}$ satisfies ${T}^{*}\psi \in {K}^{+}$ (anticone of $K$ with respect to ${X}^{\prime}$), then $\psi (y)\ge 0$.
Here ${T}^{\mathrm{*}}\mathrm{:}{Y}^{\mathrm{\prime}}\mathrm{\to}{X}^{\mathrm{\prime}}$ denotes the pullback, restricted to ${Y}^{\mathrm{\prime}}$ and ${X}^{\mathrm{\prime}}$, defined by ${T}^{\mathrm{*}}\mathit{}\psi \mathrm{=}\psi \mathrm{\circ}T$.
Proof.
Make the substitutions $X\to Y$, ${X}^{\prime}\to {Y}^{\prime}$, $x\to y$ and $K\to T(K)$ in Theorem 1. Condition (c) is a rephrasal of condition (b). ∎
References
 1 B. D. Craven and J. J. Kohila. “Generalizations^{} of Farkas’ Theorem.” SIAM Journal on Mathematical Analysis. Vol. 8, No. 6, November 1977.
 2 David Kincaid and Ward Cheney. Numerical Analysis: Mathematics of Scientific Computing, third edition. Brooks/Cole, 2002.
Title  generalized Farkas lemma 
Canonical name  GeneralizedFarkasLemma 
Date of creation  20130322 17:20:53 
Last modified on  20130322 17:20:53 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  7 
Author  stevecheng (10074) 
Entry type  Theorem 
Classification  msc 46A03 
Classification  msc 46A20 
Classification  msc 15A39 
Classification  msc 49J35 
Synonym  Farkas lemma for topological vector spaces 
Synonym  generalized Farkas theorem 
Related topic  AntiCone 
Related topic  Cone5 